It's state that comes after the fall of one civilization and before the rise of another.
Anarchy means, rule of the strongest, thereby it isn't a system that anyone would actually want, the people who advocate it usually don't realize that their actually advocating for another system such as free market capitalism (A.K.A Anarcho-capitalism).
The fact is that we have proven time and time again that we do need some kind of democratic governing in order to secure the safety and interests of the people.
The matter can be easily resolved by looking at a global chart of anarchist societies and comparing it to Marxist-Leninist societies. Whereas anarchism has covered nothing but atom-like specks on the earth, Marxism-Leninism has spread across continents, countries, nationalities, oceans, and even space! The logical answer to this is that Marxism has a better theory, thus, puts out better practice. Anarchism is, also, not only ineffective but highly illogical. The state is a product of the means of material exchange in society. When one controls the economic means of production, one controls the intellectual means of production. The state is a tool used by the capitalist class to enforce its class power onto the people. Anarchists declare that the state must be abolished before the economic relationships are abolished. What ignorance! The economic relationships must be changed (in this case, in favor of the workers) in order to dissolve the state.
While I can see the anarchist philosophy gaining more ground in individual countries with certain political situations, it's not conducive to a world-wide revolution. Anarchism places extreme emphasis on the individual, and his/her freedom from any type of authority/coercion, even that which is arguably justified. An "every man for himself" ideology might be able to mobilize the force necessary to overthrow one government, but a successful global revolution demands cooperation, smooth logistics, and inevitably, some degree of structure. The ensuing new order would also require a degree of governmental structure to survive, at least in the beginning. As jakehmw astutely pointed out, this explains the historical track records of both ideologies. Marxism addresses the practical needs inherent in revolution, anarchism does not.